Bistable Isotropic Material

Constitutive Law

This MPM material is two small-strains materials (each with same constitutive law as an isotropic material). The two material states are linked by a transition rule and the transition between the two states can be reversible or irreversible.

The transition between the two states is determined either by a "dilation" rule, a "distortion" rule, or a "Von Mises" stress rule (as determined by the transition property). With a dilation rule, the transition occurs when the volumetric strain

      [math]\displaystyle{ {\Delta V\over V} = \varepsilon_{xx}+\varepsilon_{yy}+\varepsilon_{zz} }[/math]

reaches the entered critical value. WIth a distortion rule, the transition occurs when the second strain invariant:

      [math]\displaystyle{ I_2 = \sqrt{{1\over 2}\sum_{i,j}\varepsilon_{ij}'\varepsilon_{ij}'} }[/math]

where [math]\displaystyle{ \varepsilon_{ij}' }[/math] is deviatoric strain tensor, reaches a critical value. By a Von Mises stress rule, the transition occurs when the Von Mises stress

      [math]\displaystyle{ \sigma_{VM} = \sqrt{\sum_{i,j}\sigma_{ij}'\sigma_{ij}'} }[/math]

where [math]\displaystyle{ \sigma_{ij}' }[/math] is is deviatoric stress, reaches a critical value. When using a dilation rule, the new stress-strain relation can include a changed offset in volumetric strain corresponding to stress-free conditions at a non-zero dilation relative to the initial state (see DeltaVOffset property). This change normally leads to an instantaneous change in stress upon transition. When using a distortion or Von Mises stress rule, the offset is ignored and the change is only a change in slope of mechanical properties.

Material Properties

The material properties for each state and the transition rules are set using:

Property	Description	Units	Default
K0	Bulk modulus for initial state	MPa	none
G0	Shear modulus for initial state	MPa	none
alpha0	Thermal expansion coefficient for initial state	ppm/K	40
kCond0	Thermal conductivity for initial state	W/(m-K)	0
beta0	Solvent expansion coefficien for initial statet	1/(wt fraction)	0
D0	Solvent diffusion constant for initial state	mm2/sec	0
Kd	Bulk modulus for transformed state	MPa	none
Gd	Shear modulus for transformed state	MPa	none
alphad	Thermal expansion coefficient for transformed state	ppm/K	40
kCondd	Thermal conductivity for transformed state	W/(m-K)	0
betad	Solvent expansion coefficien for transformed statet	1/(wt fraction)	0
Dd	Solvent diffusion constant for transformed state	mm2/sec	0
transition	Set to "dilation" (or 1), "distortion" (or 2), or "vonmises" (or 3) to select the transition model	none	dilation
critical	The critical volumetric strain to induce a dilation transition (in precent strain), critical strain invariant to induce a distortion transition (in percent strain), or critical Von Mises stress to induce a vonmises transition (in MPa).	varies	none
DeltaVOffset	An offset volumetric strain in the transformed state. This property only applies for dilation rule.	%	0
reversible	Make the transition reversible (if "yes" or 1) or irreversible (if "no" or 0).	none	yes
(other)	Properties common to all materials	varies	varies

History Variables

History variable 1 will be 0 for the initial state and 1 for the deformed state after a transition. This variable can be archived as history1.